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\noindent{\Large{\textsc{Complex Variables\hfill Assignment 10  \\ Spring 2020 \hfill Due April 15}}}

\large

\vspace{1cm}

\begin{exercise}
Textbook exercise IV.4.1.
\end{exercise}

\begin{exercise}
Prove that $\int_0^{\pi} e^{\cos \theta} \cos(\sin \theta) \, d \theta = \pi$ by considering $\int_{\gamma} (e^z/z) \, dz$, where $\gamma$ is the unit circle.
\end{exercise}

\begin{exercise}
Textbook exercise IV.4.2.
\end{exercise}

\begin{exercise}
Let $f$ be analytic on a disk $D$ and let $\gamma$ be any simple loop in $D$.  Prove that
\[
\int_{\gamma} \frac{f(z)}{(z-z_0)^2} \, dz = \int_{\gamma} \frac{f'(z)}{(z-z_0)} \, dz
\]
for any $z_0 \in D \setminus \gamma$.  Generalize.
\end{exercise}

\begin{exercise}
Suppose that $f$ is analytic on $|z|<1$ and that it satisfies the inequality $|f(z)| \le 1$.  Prove that $|f'(0)| \le 1$.
\end{exercise}


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